Guys that is what Ned and his wife said. That may or may not be the truth.
Ned is a very very honorable man.
He was sworn to Catelyn.
He went to the tower where Lyanna was, killed three Kingsguard who were guarding it, and Lyanna was dying when he got there (childbirth?
Rhaegar supposedly "kidnapped Lyanna" (a lady similar in temperament to Arya meaning very strong willed and unlikely to come quietly) after declaring her the Queen of Love and Beauty at a tourney.
The Kingsguard guard the royal family, NOT their consorts.
Ned came back from the war with Jon, raises him as his son, but refuses to give many details about the mother and how it happened. He tells Robert it was Wylla, although the King is too much of a drunk to remember much. He tells his very disappointed wife the same.
Lyanna, as Ned's sister, would pass on all the Stark traits Ned would, thus making Jon closely related to Ned, if not his son. But definitely "his blood".
Thus it is not a far stretch to say that Lyanna fell for Rhaegar, seeing as how everyone loved him, they eloped, and while in the tower, she becomes fatally ill from childbirth. Ned, getting there in time to see her before she dies, promises to raise the child as his own, thus trumping his commitment to Catelyn and preserving his strong sense of honor and dignity.